A man cover a certain distance on foot and comeback by car he takes 6 hour 30 min. If he travels both side distance by car then he saves 2 hour 10 minutes. If he covers both side distance on foot then find the time taken by him.

Options

A. 7 hr, 50 min
B. 7 hr, 30 min
C. 8 hr, 40 min
D. 8 hr, 30 min

Answer

To solve the problem step by step, we will denote the time taken by foot as \( t_f \) and the time taken by car as \( t_c \).

### Step 1: Understand the total time taken
The man takes a total of 6 hours and 30 minutes to cover a certain distance on foot and return by car. We convert this time into hours:
\[ 6 \text{ hours } 30 \text{ minutes} = 6.5 \text{ hours} \]
Thus, we can write the equation:
\[ t_f + t_c = 6.5 \quad \text{(1)} \]

### Step 2: Understand the time saved when traveling both ways by car
If he travels both ways by car, he saves 2 hours and 10 minutes. We convert this time into hours:
\[ 2 \text{ hours } 10 \text{ minutes} = 2 + \frac{10}{60} = 2 + \frac{1}{6} = \frac{13}{6} \text{ hours} \]
This means that if he travels both ways by car, the time taken would be:
\[ t_c + t_c = 2t_c \]
The equation for the time saved can be written as:
\[ t_f + t_c - 2t_c = \frac{13}{6} \]
This simplifies to:
\[ t_f - t_c = \frac{13}{6} \quad \text{(2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( t_f + t_c = 6.5 \) (1)
2. \( t_f - t_c = \frac{13}{6} \) (2)

We can solve these equations simultaneously. First, we can add equations (1) and (2):
\[ (t_f + t_c) + (t_f - t_c) = 6.5 + \frac{13}{6} \]
This simplifies to:
\[ 2t_f = 6.5 + \frac{13}{6} \]

### Step 4: Convert 6.5 to a fraction
Convert 6.5 to a fraction:
\[ 6.5 = \frac{13}{2} \]
Now, we can find a common denominator to add:
\[ \frac{13}{2} = \frac{39}{6} \]
Now we can add:
\[ 2t_f = \frac{39}{6} + \frac{13}{6} = \frac{52}{6} = \frac{26}{3} \]
Thus,
\[ t_f = \frac{26}{6} = \frac{13}{3} \text{ hours} \]

### Step 5: Find \( t_c \)
Now we can find \( t_c \) using equation (1):
\[ t_f + t_c = 6.5 \]
Substituting \( t_f \):
\[ \frac{13}{3} + t_c = 6.5 \]
Convert 6.5 to a fraction:
\[ t_c = 6.5 - \frac{13}{3} = \frac{39}{6} - \frac{26}{6} = \frac{13}{6} \text{ hours} \]

### Step 6: Calculate total time if both sides are traveled by foot
If he travels both ways by foot, the total time taken would be:
\[ \text{Total time} = t_f + t_f = 2t_f = 2 \times \frac{13}{3} = \frac{26}{3} \text{ hours} \]
Convert this to hours and minutes:
\[ \frac{26}{3} \text{ hours} = 8 \text{ hours } 40 \text{ minutes} \]

### Final Answer
The time taken by him if he covers both sides distance on foot is **8 hours 40 minutes**.